mecanica_lagrangiana.pdf

7/21/2019 Mecanica_Lagrangiana.pdf

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2/45


3/45

y= f(x)

y x z= F[f]

z f(x)

z= F[f] =

x2x1

dx [f(x)]2 .

f(x) [x1, x2]

y

xy

f(x1) =y1 f(x2) =y2

F[f] =

x2x1

dxg (f(x), f(x); x) ,

f(x) F

F

(x1, y1) (x2, y2)

f(x1) =y1, f(x2) =y2.


4/45

(x1, y1) (x2, y2)

f(x)

xy

y= f(x)

F[f] =

21

dl.

dl=

dx2 +dy2 =dx

1 + (dy/dx)2 =dx

1 + [f(x)]2

F[f] =

x2x1

dx

1 + [f(x)]2,

f(x1) =y1

f(x2) =y2

f(x)

f(x) =y1+ x x1x2 x1 (y2 y1),

f(x)


5/45

[x1, x2] f(x)

f(x1) =f(x2) = 0 F

f

F =

x2x1

dxg (f(x), f(x); x) =

x2x1

dxg (f(x), f(x); x) =

x2x1

dx

g

ff+

g

ff

.

f =(df/dx) = (d/dx)f x2x1

dxg

ff =

x2x1

dxg

fd

dxf=

g

ff(x)

x2

x1

x2x1

dxd

dx

g

f

f.

F =

x2x1

dx

g

f d

dx

g

f

f.

f(x) F = 0

g

f d

dx

g

f

= 0.

f(x)

F = 0

F = 0

2F

2F

f(x) x

df = 0 df

dx

df= df

dxdx= 0 df

dx= 0,

dx = 0


6/45

d2f= 12d2f

dx2dx2

g(f, f

; x) =

1 +f

2 g

f d

dx g

f

= 0 d

dx f

1 +f2

= 0,

f1 +f2

=c f = c1 c2 a,

c a

f(x) = dxa= ax+b. a b

g

x


7/45

xy

dl x

y dt= dl/v v

x

x= 0 0 = 12

mv2mgx v(x) = 2gx g g(f, f; x)

F[f] =

21

dl

v =

x2x1

dx

1 +f2

2gx .

g

f d

dx

g

f

= 0 g(f, f; x) =

1 +f2

2gx ,

f(x1) =y1

f(x2) =y2

g/f= 0

g

f=

f2gx(1 +f2)

=c,

c f

f2

2gx(1 +f2)=c2 f =

x

2a x, 2a 1

2gc2.

x

f(x) =

xdx

2ax x2 .

x= a(1 cos )

f(x) =

a2(1 cos )sin d2a2(1 cos ) a2(1 cos ) =a

d(1 cos ) =a( sin ) +b.

x= a(1 cos )y= a( sin ) +b

a, b x1 = y1 = 0

1 = 0 b = 0 a

y (x, y)


8/45

xy xy a

y

O

a

g

x d

dx

g fg

f

= 0

g x g/x = 0

g fgf

= 0.

R (1, 1) (2, 2)

dl= R

d2 + sin2 d2 =Rd

sin2 +

d

d

2.

d


9/45

()

F[] =R

21

d

sin2 +2,

(1) =1(2) =2

g(, ; ) =R

sin2 +2 g

g g

=R

sin2 +2 R2

sin2 +2

=aR,

a R R

sin2 +2

sin2 = a

sin2 +2.

d

d =

a

sin

sin2 a2

=

ad

sin

sin2 a2

.

=

+, =

1 a2a2

.

= sin ( ) =sin cos cos sin .


10/45

R sin

R cos = B(R sin cos ) A (R sin sin ) ,

A = cos B = sin

Bx Ay= z,

A B

cos 1 = Bsin 1cos 1 A sin 1sin 1, cos 2= Bsin 2cos 2 A sin 2sin 2.

f1(x), f2(x), , fN(x) {fi(x)}

F[{fi}] = x2x1

dxg(f1, f

1, f2, f

2, , fN, fN; x),

fi(x1) = f(1)i fi(x2) = f

(2)i

N

g

fi d

dt

g

fi

= 0, i= 1, 2, , N.

F fi

[x1, x2]

F =N

i=1 x2x1

dx

g

fi d

dt

g

fi

fi(x) = 0,

fi

f(x, y)

(x, y) f

df=f

xdx+

f

ydy= 0.


11/45

dx dy

df= 0

f

x = 0,

f

y = 0.

fi(x)

f1 f2 F[f1, f2]

G(f1, f2) = 0

G(f1, f2) = 0

f2 = H(f1) g(f1, f

1, f2, f

2; x)

g(f1, f

1, f2, f

2; x) =g(f1, f

1, H(f1), H(f1); x) = g(f1, f

1; x).

f2= H(f1)f

1 F[f1, f2] = F[f1]

F[f1, f2] = F[f1] =

x2x1

dxg(f1, f

1, f2, f

2; x) =

x2x1

dxg(f1, f

1; x).

g

f1 d

dt

g

f1

= 0.

f1(x) f2(x) = H(f1(x))

f2 f1

F =

x2x1

dx

g

f1 d

dx

g

f1

f1(x) +

g

f2 d

dx

g

f2

f2(x)

= 0.

f1 f2

G = G

f1f1(x) +

G

f2f2(x) = 0,


12/45

f2(x)

f2(x) = G/f1G/f2

f1(x).

F= x2x1 dx

gf1

d

dx g

f1 gf2 ddx gf2

G/f1

G/f2

f1(x) = 0.

f1(x)

g

f1 d

dx

g

f1

G

f1

1

=

g

f2 d

dx

g

f2

G

f2

1

.

x

f1(x) f2(x)

(x) f1(x) f2(x) (x)

g

f1 d

dx

g

f1

+(x)

G

f1= 0,

g

f2 d

dx

g

f2

+(x)

G

f2= 0,

G(f1, f2) = 0.

N

fi(x)

i= 1, , N M Gj({fi}) = 0 j= 1, , M M < N

g

fi d

dx

g

fi

+

Mj=1

j(x)Gjfi

= 0, i= 1, , N

Gj({fi}) = 0, j = 1, , M

N+ M N+ M {fi(x)} j(x)

F[{fi}] =F[{fi}] +Mj=1

x2x1

dxj(x)Gj({fi}),


13/45

{fi} {j(x)}

F[{fi}] = F[{fi}] +Mj=1

x2x1

dxj(x)Gj({fi})

= x2

x1

dxN

i=1 g

fi

d

dx g

f

i fi(x)

+

x2x1

dxMj=1

Gj({fi})j(x) +j(x)

Ni=1

G

fifi(x)

.

F[{fi}] = x2x1

dx

Ni=1

g

fi d

dx

g

fi

+

Mj=1

j(x)G

fi

fi(x)

+Mj=1

Gj({fi})j(x)

.

fi j

f(x, y)

h(x, y) = 0 dx dy

dh=h

xdx+

h

ydy dy= h/x

h/ydx

dy dx

df=f

xdx+

f

ydy=

f

x f

y

h/x

h/y

dx= 0.

dx

f

x

h

x

1

=f

y

h

y

1

.

(x, y)

f

x+

h

x= 0,

f

y +

h

y = 0, h(x, y) = 0,


14/45

x, y

f

x= 0,

f

y = 0,

f

= 0,

f(x, y) f(x, y) +h(x, y)

F[f] =

21

dt=

21

ds

v =

1

c

21

nds,

n c xy

n n(x, y)

F =1

c

x2x1

dxn (x, f(x))

1 + [f(x)]2 = 0.

g(f, f; x) =n (x, f)

1 +f2

N xi = 1, , N i = 1, 2, 3 T = Ec U

S[{xi}] = t2t1

dtL ({xi, xi}) , L ({xi, xi}) T({xi}) U({xi}) ,

L S

t1< t2

xi(t1) =x(1)i, xi(t2) =x

(2)i

{x(1)i, x(2)i } xi(t)

S= 0.


15/45

L

xi d

dt

L

xi

= 0,

xi(t1) =x

(1)i

xi(t2) =x(2)i

= 1, , Ni= 1, 2, 3.

m

k

L=1

2mx2 1

2kx2.

L

x =

kx,

d

dtL

x = d

dt(mx) =mx,

mx+km = 0,

m

l T = 12ms2

s

s = l T = 12ml22


16/45

U=mgl(1cos )

L=1

2ml22 mgl(1 cos ).

L

d

dt

L

= 0.

L

= mgl sin , d

dt

L

=

d

dt

ml2

=ml2,

mgl sin +ml2= 0 + gl sin = 0.

N

s

s= 3N M M s

q1(t), q2(t), , qs(t) = {qj(t)}.


17/45

q1(t), q2(t), , qs(t) = {qj(t)}.

xi= xi(q1, q2, , qs; t) =xi({qj}; t), xi= xi({qj, qj}; t).

qj =qj({xi}; t), qj = qj({xi, xi}; t).

S[{qj}] = t2t1

dtL ({qj, qj}; t) = 0,

L

L ({xi, xi}) = T({xi}) U({xi})= T({qj, qj}; t) U({qj}; t) L ({qj, qj}; t) .

L

qj d

dt L

qj

= 0, j = 1, , s.

s= 1

N = 1

M = 2 s = 3 1 2 = 1 q


18/45

q=

x= l sin , y= l cos ,

O

x

x= l cos , y= l sin ,

L=1

2m

x2 + y2

=1

2ml22.

U=mgy = mgl cos .

L= T U=12

ml22 +mgl cos ,

L

d

dtL

= mgl sin d

dt

ml2

= mgl sin ml2= 0,

+ glsin = 0

(x, y)

L=1

2m

x2 + y2 mgy.

L/x= 0 x

L

y =mx=


19/45

x y

L

y d

dt

L

y

= mg my= 0 y= g.

2 +g sin = 0,

2+ 2+g cos = 0.

L({qj , qj}) =T({qj}) U({qj}) L

qj d

dt

L

qj

= U

qj d

dt

T

qj

Qj

Qj = Uqj

T

mqj

mqj =Qj ,

U=U({qj}; t) T

qj U

qj d

dt

T

qj

= 0,

d

dt

T

qj

T

qj=Qj

Qj

Qj Uqj

.


20/45

m

s = 2 (,,z)

= ztan 2 z

x= cos

y= sin

z= 2

x= cos sin y= sin + cos

z= 2

T =1

2

m x2 + y2 + z2 =12

m2 +22 + 2 2 2 .

1 + 2

2 = 1 +

1

tan2 2=

1

sin2 2= 2

2,

T

T =1

2m

2 2

2+22

.


21/45

U=mgz= mg

2,

L=

1

2m

2

2

2 +2

2 mg 2 .

L/= 0

L

=m2= lz

z

L = mr v= m (e+zez) ( e+e+ zez)

=z e+m(z z)e+m2

ez

=2 2

e+m2ez.

L

d

dt

L

=m2 mg

2 m 2

2 = 0.

lz

l2zsin

2 2

m23 +

1

2g sin = 0.

a= s= 1

q=

x= 12

at2 +l sin

y= l cos

t0 = 0 O

x0 = 0 v0 = 0 x= at+l

cos

y= l sin


22/45

t

L= T U=12

m

x2 + y2 mgy = 1

2m

at+l cos 2

+1

2m2l2 sin2

+mgl cos =1

2ma2t2 +

1

2ml22 +matl cos +mgl cos .

L

d

dt

L

=matl sin mgl sin d

dt

ml2+matl cos

= matl sin mgl sin ml2 mal cos +matl sin = 0.

+a

lcos +

g

lsin = 0.

a= 0


23/45

= 0

e

a

lcos e+

g

l sin e= 0 tan e= a

g.

a > 0 a = 0

e

= e || 1

+a

lcos(+e) +

g

lsin(+e)

= +a

l(cos cos e sin sin e) + g

l(sin cos e+ cos sin e)

= + al

cos e+g

l

sin e cos + a

l

sin e+g

l

cos e sin = 0, sin

+a

l sin e g

lcos e

= 0.

a cos e+g sin e= 0

cos2 e+ sin2 e= 1

cos e= g

g2 +a2, sin e= a

g2 +a2

e < 0

a

l sin e g

lcos e =

g2 +a2

l ,

+2= 0, 2 =

g2 +a2

l .

g/l a= 0

a = 0


24/45

M

M

l1 l2

O

x

q1 m1 q2

m2 xi i= 1, 2, 3

x1= q1, x2 = l1 q1+q2, x3 = l1 q1+l2 q2.

x1 = q1, x2 = q1+ q2, x3= q1 q2.

T =1

2m1q

21+

1

2m2(q1 q2)2 +1

2m3(q1+ q2)

2 ,

U= m1gq1 m2g (l1 q1+q2) m3g (l1 q1+l2 q2) ,


25/45

T = 1

2m1q

21+

1

2m2(q1 q2)2 +1

2m3(q1+ q2)

2

+ m1gq1+m2g (l1 q1+q2) +m3g (l1 q1+l2 q2) .

Lq1

ddt

Lq1

= (m1 m2 m3)g (m1+m2+m3)q1+ (m2 m3)q2= 0,

L

q2 d

dt

L

q2

= (m2 m3)g+ (m2 m3)q1 (m2+m3)q2 = 0.

(m1+m2+m3)q1+ (m3 m2)q2= (m1 m2 m3)g,(m2 m3)q1 (m2+m3)q2= (m3 m2)g,

q1 q2

fi(q1, q2, ; t) = 0, i= 1, 2, , M


26/45

M s = 3N M s {qj} j = 1, 2, , s s

L

qj d

dt

L

qj

= 0, j= 1, 2, , s,

L

qj d

dt

L

qj

+

Mi=1

i(t)fiqj

= 0, j= 1, 2, , s.

L({qj, qj}) =T({qj}) U({qj})

L

qj d

dt L

qj+

M

i=1

i(t)fi

qj=

U

qj d

dtT

qj+

M

i=1

i(t)fi

qj

= Fj mqj+Mi=1

i(t)fiqj

= 0.

mqj =Fj+Mi=1

i(t)fiqj

.

QjMi=1

Qij , Qij i(t) fiqj

Qj j Qij i

j

Qij


27/45

R M

x y

q1= x

q2 =

y = 0 x= 0

f(x, ) =x R = 0.

T =1

2Mx2 +

1

2I2,

I = MR2/2

U= Mgx sin

L=1

2Mx2 +

1

2I2 +M gx sin .

L

x d

dt

L

y

+(t)

f

x= 0 Mg sin Mx+= 0,

L

d

dtL

+(t)

f

= 0 1

2MR2 R= 0.

x(t) (t) (t) = x/R

1

2MR2

x

R

+R= 0 = 1

2Mx.


28/45

Mg sin Mx 12

Mx= 0 x= 23

g sin .

= 13

Mg sin , = 2g

3Rsin .

t0 = 0 x0 = 0

0= x0= 0 0 = 0

x(t) =1

3gt2 sin , (t) =

1

3Rgt2 sin .

2/3

g sin = 90

I = MR2

x= g2sin = g2Rsin


29/45

Qx = f

x= = 1

3Mg sin ,

Q = f

= R= R

3Mg sin .

Qx< 0

Q >0

= x/R x x

L=1

2mx2 +

1

2I

x

R

2+M gx sin =

3

4Mx2 +Mgx sin .

L

x d

dt

L

x

=M g sin 3

2Mx= 0 x= 2

3Mg sin .

fi(q1, q2, ) = 0, i= 1, 2, , M

fi({qj , qj}; t) = 0, i= 1, 2, , M,

j

Ajqj+B = 0,


30/45

h({qj}; t)

Aj = h

qj, B=

h

t,

j

Ajqj+B =j

h

qj qj+

h

t =dh

dt = 0 h({qj}; t) c= 0,

c

x R

x x

R= 0

f(x, x,, ) = x R= ddt

(x R) = 0 x R= c.

c y y = 0

= 0 c= 0

T =1

2

i

mx2i.

xi= xi({qj}; t)

xi= j

xi

qjqj+

xi

t .

x2i=kj

xiqj

xiqk

qjqk+ 2j

xiqj

xit

qj+

xi

t

2,


31/45

T =ijk

1

2m

xiqj

xiqk

qjqk+ij

mxiqj

xit

qj+i

1

2m

xi

t

2

=

jkajkqjqk+ j

bjqj+ c,

ajk =akj 12

i

mxiqj

xiqk

, bji

mxiqj

xit

, c=1

2

i

m

xi

t

2.

xi= xi({qj})

T = jk

ajkqjqk.

T

ql=k

alkqk+j

ajlqj ,

l

qlT

ql=kl

alkqkql+kl

ajlqjql = 2kl

alkqkql= 2T,

l

qlT

ql= 2T

f(y1, y2, , yN) n

f(y1, y2, , yN) =nf(y1, y2, , yN),

Ni=1

yif

yi=nf

i

yif(y1, , yN)

(yi)

=1

=n n1f=1

= 1


32/45

L= L({qj, qj}; t)

t0

L({qj, qj}; t) =L({qj, qj}; t+t0) Lt

= 0.

L

dL

dt =j

L

qjqj+

j

L

qjqj+

L

t =j

L

qjqj+

j

L

qjqj.

dL

dt =j

d

dt

L

qj

qj+

j

L

qjqj =

j

d

dt

qj

L

qj

.

d

dt

j

qjL

qj L

= 0 H

j

qjL

qj L=

H

U U=U({qj})

L

qj=

T

qj


33/45

H=j

qjL

qj L=

j

qjT

qj T+U.

U=U({qj}) xi= xi({qj})

H= 2T T+U=T+U=E,

E

H

L H=E E

L H= E

U H = E

E L

H=E

H

qk

L(q1, , qk, , qs; t) =L(q1, , qk+c, , qs; t) Lqk

= 0,


34/45

c L

qk L

qk qk

d

dt L

qk

= 0 pk L

qk =

pk

pk =pk(q1, , qk1, qk+1, , qs, q1, , qs; t) =

qk pk

qk pk

N

Fi = 0 ri

i


35/45

i

Fi ri = 0 Fi= pi

Fi pi = 0 i

(Fi pi) ri= 0 Fi fi

i

(Fi+ fi pi) ri= i

Fi ri+ i

fi ri i

pi ri= 0.

i

Fi ri i

pi ri = 0

ri

ri = ri({qj}; t)

{qj}

j = 1, 2, , s

ri =j

riqj

qj ,

i

Fi ri i

pi ri =ij

Fi riqj

qj ij

miri riqj

qj = 0.

miri riqj

= d

dt

miri ri

qj

miri d

dt

riqj

.

d

dt

riqj

=

riqj

,

ri

qj=

ri

qj.

d

dt

riqj

=k

2riqkqj

qk+ 2riqjt

,

riqj

=

qj

k

riqk

qk+ri

t

=k

2riqkqj

qk+ 2riqjt

,


36/45

riqj

=

qj

k

riqk

qk+ri

t

=

riqj

.

Qj =iFi

riqj ,

j

Qj

i

d

dt

miri ri

qj

miri ri

qj

qj = 0.

i

miri riqj

=

qj

1

2 i

mi |ri|2

= T

qj,

i

miri riqj

=

qj

1

2

i

mi |ri|2

= T

qj.

j

Qj

d

dt

T

qj

T

qj

qj = 0.

qj

d

dt

T

qj

T

qj=Qj, j = 1, 2, , s.

U

Fi =

riU

Qj =i

Fi riqj

= i

riUriqj

= Uqj

.

U {qj} d

dt

T

qj

T

qj=

d

dt

L

qj

T

qj= U

qj


37/45

d

dt

L

qj

T

qj+

U

qj=

d

dt

L

qj

L

qj= 0,

S[{qj}] = t2t1

dt (T+V) = 0,

V

V =i

Fi ri.

S =

t2t1

dtT+

t2t1

dti

Fi ri = t2t1

dtT+

t2t1

dti

Fi ri

=

t2t1

dtT+

t2t1

dtj

Qjqj.

j Q

jqj =ijFi

ri

qj qj =iFi j

ri

qj qj =iFi ri.

T =T({qj, qj}; t)

t2t1

dtT =

t2t1

dtj

T

qj d

dt

T

qj

qj.

t2t1

dtj

Tqj

ddt

Tqj

+Qj

qj = 0

qj


38/45

{qj} {qj} L= L({qj, qj}; t) {qj, pj}

H=j

qjL

qj L=

j

qjpj L.

pj = L/qj L {qj, qj}

{qj , pj}

pj = L

qj=pj(

{qj , qj

}; t)

qj = qj(

{qj , pj

}; t).

H({qj , pj}; t) =j

qjpj L=j

qj({qj , pj}; t)pj L({qj, qj({qk, pk}; t)}; t).

H

{qj , pj}

L= L({qj , qj}; t) H=H({qj, pj}; t)

y = f(x) x p = f(x)

f(x) p


39/45

x y

y = f(x)

(0, g)

y = g+ px g = y px y f(x) g(p) = f(x) px f(x) g(p) p=f(x) px

L {qj} H {pj} L qjpj H

y = f(x)

p g

(x, y) (g, p) g

y p

H

H

dH=j

H

qjdqj+

H

pjdpj

+

H

tdt.


40/45

dH =j

(pjdqj+ qjdpj) j

L

qjdqj+

L

qjdqj

L

t

= j (pjdqj+ qjdpj) j ( pjdqj+pjdqj) L

tdt.

L/qj = pj

pj =L/qj

j

H

qjdqj+

H

pjdpj

+

H

tdt=

j

(qjdpj pjdqj) Lt

dt

qj =H

pj, pj =

H

pj, j = 1, 2,

, s.

H

t =L

t

2s

s

2s

{qj0, pj0} 2s

{qj0, qj0

} (qj, pj)

(qj, qj)

dH=j

H

qjdqj+

H

pjdpj

+

H

tdt=

j

( pjdqj+ qjdpj) + Ht

dt=H

tdt,

dHdt

= Ht

.

H H=

H L/t = 0

H/t= L/t


41/45

{qj} j = 1, 2, , s

L= T U

pj =L/qj

qj = qj({qj, pj}; t)

H= jpjqjL

H

R

r

m

x2 +y2 =2 =R2 F = kr = R

q1 = q2 = z U(r)


42/45

rU= Ur

er+1

r

U

e+

1

r sin

U

e= krer.

U

e= 0 U=U(r, ),

U

e = 0 U=U(r),

U

r =kr U(r) =k

r0

drr =1

2kr2 =

1

2k

R2 +z2

,

r= 0

x= cos = R cos , y= sin = R sin , z= z

x= R sin , y= R cos , z= z,

L= T U = 12

m

x2 + y2 + z2 1

2k

R2 +z2

= 1

2m

R22 + z21

2k

R2 +z2

.

p=L

=mR2, pz =

L

z =mz.

= p

mR2, z=

pz

m.

H = p+ zpz L=p2

mR2+

p2zm

12

m

R2 p

mR2

2+pz

m

2+

1

2k

R2 +z2

=p2

2mR2+

p2z2m

+1

2k

R2 +z2

.


43/45

L H=E= T+ U= H=T+ U

+ T+U

p= H

= 0 (1) pz = Hz

= kz (3)

= H

p=

pmR2

(2) z=H

pz=

pzm

(4)

p= mR2=

z (t) =pt/mR2 + c c

z= pzm

= km

z z+20z= 0,

=

k/m z

H

qj H L

qj L

L= L(q1, q2, , qj1, qj+1, , qs; q1, q2, , qj1, qj , qj+1, , qs; t),


44/45

s 1 pj =L/qj 2(s 1) + 1 = 2s 1 qj

pj = 0 pj =j = ,

qj =H

j(q1, q2, , qj1, qj+1, , qs;p1, p2, , pj1, , pj+1, , ps; t).

qj

2s2 2s2 qj

qj(t) =

dtj(q1, q2, , qj1, qj+1, , qs;p1, p2, , pj1, , pj+1, , ps; t).

s

pj =j =

qj(t) =

dtj =jt+j,

j


45/45

O

mr +mR= F,

R O

O R(t) = 12at2ex

R= aex

F =Fg+ T = mgey Te,

T

ml2

e+mle= maex mgey Te

=

ma(sin e+ cos e)

mg(

cos e+ sin e)

Te

= 0 = 0 = + 34

ml

2 = ma sin +mg cos Tml= ma cos mg sin

T

pi

qj

= 0

mecanica_lagrangiana.pdf

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